1. Lecture 11: Networks II: conductance-based synapses, visual cortical hypercolumn model References: Hertz, Lerchner, Ahmadi, q-bio.NC/0402023 [Erice lectures] Lerchner, Ahmadi, Hertz, q-bio.NC/0402026 (Neurocomputing, 2004) [conductance-based synapses] Lerchner, Sterner, Hertz, Ahmadi, q-bio.NC/0403037 [orientation hypercolumn model]

2. Conductance-based synapses In previous model:

3. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance:

4. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance:

5. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train

6. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

7. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

8. Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

9. Model

12. Mean field theory Effective single-neuron problem with synaptic input current

13. Mean field theory Effective single-neuron problem with synaptic input current

14. Mean field theory Effective single-neuron problem with synaptic input current with

15. Mean field theory Effective single-neuron problem with synaptic input current with where = correlation function of synaptically-filtered presynaptic spike trains

16. Balance condition Total mean current = 0 :

17. Balance condition Total mean current = 0 :

18. Balance condition Total mean current = 0 : Mean membrane potential just below 

19. Balance condition define Total mean current = 0 : Mean membrane potential just below 

20. Balance condition define Total mean current = 0 : Mean membrane potential just below 

21. Balance condition define Solve for r b as in current-based case: Total mean current = 0 : Mean membrane potential just below 

22. Balance condition define Solve for r b as in current-based case: Total mean current = 0 : Mean membrane potential just below 

23. Balance condition define Solve for r b as in current-based case:  Total mean current = 0 : Mean membrane potential just below 

24. High-conductance-state

26. V a “chases” V s a (t) at rate g tot (t)

27. High-conductance-state V a “chases” V s a (t) at rate g tot (t)

28. High-conductance-state V a “chases” V s a (t) at rate g tot (t)

29. High-conductance-state V a “chases” V s a (t) at rate g tot (t) Effective membrane time constant ~ 1 ms

30. Membrane potential and spiking dynamics for large g tot

31. Fluctuations Measure membrane potential from :

32. Fluctuations Measure membrane potential from :

33. Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

34. Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

35. Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

36. Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations:

37. Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: =>

38. Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or

39. Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or with

40. Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or with

41. Effective current-based model High connectivity:

42. Effective current-based model High connectivity:

43. Effective current-based model High connectivity:

44. Effective current-based model High connectivity:

45. Effective current-based model High connectivity:

46. Effective current-based model High connectivity: Like current-based model with

47. Effective current-based model High connectivity: Like current-based model with (but effective membrane time constant depends on presynaptic rates)

48. Firing irregularity depends on reset level and  s

49. Modeling primary visual cortex

50. Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings)

51. Modeling primary visual cortex Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings)

52. Modeling primary visual cortex Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings) Note: contrast- invariant tuning width

53. Spatial organization of area V1 2. In V1, nearby neurons have similar orientation tuning

54. Spatial organization of area V1 2. In V1, nearby neurons have similar orientation tuning

55. Orientation column ~ 10 4 neurons that respond most strongly to a particular orientation

56. Orientation column ~ 10 4 neurons that respond most strongly to a particular orientation Tuning of input from LGN (Hubel-Wiesel):

57. Hubel-Wiesel feedforward connectivity cannot by itself explain contrast-invariant tuning Simplest model: cortical neurons sums H-W inputs, firing rate is threshold-linear function of sum

58. Hubel-Wiesel feedforward connectivity cannot by itself explain contrast-invariant tuning Simplest model: cortical neurons sums H-W inputs, firing rate is threshold-linear function of sum

59. Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

60. Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

61. Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

62. Modeling a “hypercolumn” (2)

64.  0 is stimulus orientation

65. Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

66. Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

67. Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

68. Modeling a “hypercolumn” (2)  0 is stimulus orientation Connection probability falls off with increasing  ’, reflecting probable greater distance. (simplest model periodic in  with period  )

69. Mean field theory Effective intracortical input current

70. Mean field theory Effective intracortical input current mean

71. Mean field theory Effective intracortical input current mean fluctuations:

72. Mean field theory Effective intracortical input current mean fluctuations: with

73. Mean field theory Effective intracortical input current mean fluctuations: with Solve self-consistently for order parameters

74. Balance condition Total mean current vanishes at all  :

75. Balance condition Total mean current vanishes at all  :

76. Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation:

77. Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation:

78. Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation: Integral equations for r a (  )

79. Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation: Integral equations for r a (  ) Can take  0 = 0

80. Broad tuning

81. Make ansatz

82. Broad tuning Make ansatz use

83. Broad tuning Make ansatz use

84. Broad tuning Make ansatz use => with

85. Broad tuning Make ansatz use => with Solve for Fourier components:

86. Broad tuning Make ansatz use => with Solve for Fourier components:

87. Broad tuning Make ansatz use => with Solve for Fourier components: Valid for (otherwise r a (  ) < 0 at large  )

88.  narrow tuning useonly for

89.  narrow tuning useonly for i.e.,

90.  narrow tuning useonly for i.e., same  c for both populations – consequence of

91.  narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in

92.  narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in

93.  narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in Balance condition:

94.  narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in Balance condition: =>

95. Narrow tuning (2) Now do the integrals:

96. Narrow tuning (2) Now do the integrals:

97. Narrow tuning (2) Now do the integrals: where

98. Narrow tuning (2) Now do the integrals: where f0:f2:f0:f2: ______ ----------

99. Narrow tuning (3)

100. Divide one by the other:

101. Narrow tuning (3) Divide one by the other: determines  c

102. Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments)

103. Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments) Then can solve for rate components:

104. Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments) Then can solve for rate components:

105. Noise tuning Input noise correlations:

106. Noise tuning Input noise correlations:

107. Noise tuning Input noise correlations:

108. Noise tuning Input noise correlations: =>

109. Noise tuning Input noise correlations: => Same integrals as in rate computation =>

110. Noise tuning Input noise correlations: => Same integrals as in rate computation =>

111. Noise tuning Input noise correlations: => Same integrals as in rate computation => using

112. Noise tuning Input noise correlations: => Same integrals as in rate computation => using=>

113. Noise tuning Input noise correlations: => Same integrals as in rate computation => using=>Same tuning as input!

114. Some numerical results (1)

115. Numerical results (2): Fano factor tuning

116. Numerical results (3): noise tuning vs firing tuning